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source("utils.R")
theme_set(theme_light())Lattice data analysis – multivaraite methods for HTS-based data
In this vignette we will show:
Multivariate lattice data analysis methods for HTS-based approaches.
This includes global metrics on the entire field of view and local variants thereof.
The use case is a 10X Visium data set from McKellar et al. (2021).
Complementary resources using this data and methods are found in the Voyager bivariate vignette and Voyager multivariate vignette
Dependencies
Setup and Preprocessing
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# taken from https://pachterlab.github.io/voyager/articles/visium_10x.html
#spe_vis <- readRDS("../data/spe_spot.rds")
#spe_vis
sfe_full <- SFEData::McKellarMuscleData(dataset = "full")
sfe_full <- mirrorImg(sfe_full, sample_id = "Vis5A", image_id = "lowres")
sfe <- sfe_full[,colData(sfe_full)$in_tissue]
sfe <- sfe[rowSums(counts(sfe)) > 0,]
#perform normalisation
sfe <- scater::logNormCounts(sfe)
colGraph(sfe, "visium") <- findVisiumGraph(sfe)Given this data from McKellar et al. we choose two genes to analyse henceforth, named Mdk (ENSMUSG00000027239) and Ncl (ENSMUSG00000026234) (McKellar et al. 2021).
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MdK <- "ENSMUSG00000027239"
NcI <- "ENSMUSG00000026234"Show the code
features <- c("ENSMUSG00000027239", "ENSMUSG00000026234") # MdK, Ncl
colGraphName <- "visium"
colGeometryName <- "spotPoly"
segmentation <- "spotPoly"Regular lattice and spatial weight matrix
Spot based data is collected along a regular spaced grid where all sample areas have the same size. Such a grid is also called a regular lattice. In more rigorous terms the data \(Y\) is a realisation of a random variable fixed along a lattice \(D\). The lattice \(D\) does not have to be regular but in the scope of spot based data it is. Spot-based data is generated by a defined sampling strategy, whereas point pattern data is the result of a stochastic process (Zuur, Ieno, and Smith 2007).
The lattice is composed of individual spatial units
\[D = \{A_1, A_2,...,A_n\}\]
where these units do not overlap
\[A_i \cap A_j = \emptyset \forall i \neq j\]
The data is then a random variable of the spatial unit along the lattice
\[Y_i = Y(A_i)\]
Most lattice data analysis technique build on the concept of neighbours. Therefore, the spatial relationship can be modelled with e.g. a spatial weight matrix \(W\). There are a lot of ways to define a spatial weight matrix \(W\). For example, the spatial weight matrix can be defined as a binary contiguity matrix where adjacent units are indicated by one and non-adjacent units as zero.
\[w_{ij} = \begin{cases} 1 \text{ if } A_i \text{ and } A_j \text{ are adjacent}\\ 0 \text{ otw} \end{cases}\]
other options to specify the weight matrix \(W\) are mentioned in Zuur, Ieno, and Smith (2007).
Voyager has a special function for the construction of the weight matrix in Visium data findVisiumGraph. This function defines the neigbours to be the surrounding spots.
Global Measures for Bivariate Data
Global Bivariate Moran’s \(I\)
For two continous variables the global bivariate Moran’s \(I\) is defined as (Wartenberg 1985; Bivand 2022)
\[I_B = \frac{\Sigma_i(\Sigma_j{w_{ij}y_j\times x_i})}{\Sigma_i{x_i^2}}\]
where \(x_i\) and \(y_i\) are the two variables of interest and \(w_{ij}\) is the value of the spatial weights matrix for positions \(i\) and \(j\).
The global bivariate Moran’s \(I\) is a measure of correlation between the variables \(x\) and \(y\) where \(y\) has a spatial lag. The result might overestimate the spatial autocorrelation of the variables due to the non-spatial correlation of \(x\) and \(y\) (Bivand 2022).
Implementation using spdep
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res <- spdep::moran_bv(x = logcounts(sfe)[features[1],],
y = logcounts(sfe)[features[2],],
listw = colGraph(sfe, colGraphName),
nsim = 499)
res
DATA PERMUTATION
Call:
boot(data = xx, statistic = bvm_boot, R = nsim, sim = "permutation",
listw = listw, parallel = parallel, ncpus = ncpus, cl = cl)
Bootstrap Statistics :
original bias std. error
t1* 0.01571419 -0.01606844 0.01414984Show the code
plot(res)
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ci <- boot::boot.ci(res, conf = c(0.99, 0.95, 0.9), type = "basic")
ciBOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 499 bootstrap replicates
CALL :
boot::boot.ci(boot.out = res, conf = c(0.99, 0.95, 0.9), type = "basic")
Intervals :
Level Basic
99% (-0.0049, 0.0689 )
95% ( 0.0041, 0.0585 )
90% ( 0.0087, 0.0548 )
Calculations and Intervals on Original Scale
Some basic intervals may be unstableThe value t0 indicates the test statistic of global bivariate Moran’s \(I\). The global bivariate Moran’s \(I\) value for the genes ENSMUSG00000027239, ENSMUSG00000026234 is 0.0157142. Significance can be assessed by comparing the permuted confidence interval with the test statistic.
Global Bivariate Lee’s \(L\)
Lee’s \(L\) is a bivariate measure that combines non-spatial pearson correlation with spatial autocorrelation via Moran’s \(I\) (Lee 2001). This enables us to asses the spatial dependence of two continuous variables in a single measure. The measure is defined as
\[L(x,y) = \frac{n}{\sum_{i=1}^n(\sum_{j=1}^nw_{ij})^2}\frac{\sum_{i=1}^n[\sum_{j=1}^nw_{ij}(x_j-\bar{x})](\sum_{j=1}^nw_{ij}(y_j-\bar{y}))}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2}\sqrt{\sum_{i=1}^n(y_i-\bar{y})^2}}\]
where \(w_{ij}\) is the value of the spatial weights matrix for positions \(i\) and \(j\), \(x\) and \(y\), the two variables of interest and \(\bar{x}\) and \(\bar{y}\) their means (Lee 2001; Bivand 2022).
Implementation using voyager
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res_lee <- calculateBivariate(sfe, type = "lee.mc",
feature1 = features[1], feature2 = features[2],
colGraphName = colGraphName,
nsim = 499)
res_lee$lee.mc_statisticstatistic
0.0213312 Show the code
res_lee$lee.mc_p.value[1] 0.008The effect sice of bivariate Lee’s \(L\) for the genes ENSMUSG00000027239, ENSMUSG00000026234 is 0.0213312 and the associated p-value is 0.008
Local Measures for Bivariate Data
Local Bivariate Moran’s \(I\)
Similar to the global bivariate Moran’s \(I\) statistic, there is a local analogue. The formula is given by:
\[ I_i^B = x_i\sum_jw_{ij}y_j \]
This can be interesting in the context of detection of coexpressed ligand-receptor pairs. A method that is based on local bivariate Moran’s \(I\) and tries to detect such pairs is SpatialDM (Li et al. 2023).
Implementation using voyager
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sfe_tissue <- runBivariate(sfe, type = "localmoran_bv",
feature1 = features[1], feature2 = features[2],
colGraphName = colGraphName,
nsim = 499)
plotLocalResult(sfe_tissue, "localmoran_bv",
features = localResultFeatures(sfe_tissue, "localmoran_bv"),
ncol = 2, divergent = TRUE, diverge_center = 0,
colGeometryName = colGeometryName, size = 2) 
Local Bivariate Lee’s \(L\)
Similar to the global variant of Lee’s \(L\) the local variant (Lee 2001; Bivand 2022) is defined as
\[L_i(x,y) = \frac{n(\sum_{j=1}^nw_{ij}(x_j-\bar{x}))(\sum_{j=1}^nw_{ij}(y_j-\bar{y}))}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2}\sqrt{\sum_{i=1}^n(y_i-\bar{y})^2}}\] Local Lee’s \(L\) is a measure of spatial co-expression, when the variables of interest are gene expression measurements. Unlike the gobal version, the variables are not averaged and show the local contribution to the metric. Positive values indicate colocalization, negative values indicate segregation (Lee 2001; Bivand 2022).
Implementation using voyager
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sfe_tissue <- runBivariate(sfe, type = "locallee",
feature1 = features[1], feature2 = features[2],
colGraphName = colGraphName)
plotLocalResult(sfe_tissue, "locallee",
features = localResultFeatures(sfe_tissue, "locallee"),
ncol = 2, divergent = TRUE, diverge_center = 0,
colGeometryName = colGeometryName, size = 2) 
Local Measures for Multivariate Data
Multivariate local Geary’s \(C\)
Geary’s \(C\) is a measure of spatial autocorrelation that is based on the difference between a variable and its neighbours. (Anselin 2019, 1995) defines it as
\[c_i = \sum_{j=1}^n w_{ij}(x_i-y_j)^2\]
and can be generalized to \(k\) features (in our case genes) by expanding
\[c_{k,i} = \sum_{v=1}^k c_{v,i}\]
where \(c_{v,i}\) is the local Geary’s \(C\) for the \(v\)th variable at location \(i\). The number of variables that can be used is not fixed, which makes the interpretation a bit more difficult. In general, the metric summarizes similarity in the “multivariate attribute space” (i.e. the gene expression) to its geographic neighbours. The common difficulty in these analyses is the interpretation of the mixture of similarity in the physical space and similarity in the attribute space (Anselin 2019, 1995).
Implementation using voyager
To speed up computation we will use highly variable genes.
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hvgs <- getTopHVGs(sfe, fdr.threshold = 0.01)
# Subset of the tissue
sfe_tissue <- runMultivariate(sfe, type = "localC_multi",
subset_row = hvgs,
colGraphName = colGraphName)
# Local C mutli is stored in colData so this is a workaround to plot it
plotSpatialFeature(sfe_tissue, "localC_multi", size = 2, scattermore = FALSE)
We can further plot the results of the permutation test. Significant values indicate interesting regions, but should be interpreted with care for various reasons. For example, we are looking for similarity in a combination of multiple features but the exact combination is not known. Anselin (2019) write “Overall, however, the statistic indicates a combination of the notion of distance in multi-attribute space with that of geographic neighbors. This is the essence of any spatial autocorrelation statistic. It is also the trade-off encountered in spatially constrained multivariate clustering methods (for a recent discussion, see, e.g., Grubesic, Wei, and Murray 2014).”. Multi-attribute space refers here to the highly variable genes. The problem can be summarised to where the similarity comes from, the gene expression or the physical space (Anselin 2019). The same problem is common in spatial domain detection methods.
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sfe <- runMultivariate(sfe, type = "localC_perm_multi",
subset_row = hvgs,
nsim = 100,
colGraphName= colGraphName)
# stored as spatially reduced dim; plot it in this way
spatialReducedDim(sfe, "localC_perm_multi", c(1, 11))
plotted are the effect size and the adjusted p-values in space.
Local Neighbour Match Test
This test is useful to assess the overlap of the \(k\)-nearest neighbours from physical distances (tissue space) with the \(k\)-nearest neighbours from the gene expression measurements (attribute space). \(k\)-nearest neighbour matrices are computed for both physical and attribute space. In a second step the probability of overlap between the two matrices is computed (Anselin and Li 2020).
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sf <- colGeometries(sfe)[[segmentation]]
sf <- cbind(sf, t(as.matrix(logcounts(sfe)[hvgs,])))
nbr_test <- neighbor_match_test(sf[c(hvgs)], k = 20)
sf$Probability <- nbr_test$Probability
sf$Cardinality <- nbr_test$Cardinality
p <- tm_shape(sf) + tm_fill(col = 'Cardinality')
q <- tm_shape(sf) + tm_fill(col = 'Probability')
tmap_arrange(p,q)
Cardinality is a measure of how many neighbours of the two matrices are common. Some regions show high cardinality with low probability and therefore share similarity on both attribute and physical space. In contrast to multivariate local Geary’s \(C\) this metric focuses directly on the distances and not on a weighted average. A problem of this approach is called the empty space problem which states that as the number of dimensions of the feature sets increase, the empty space between observations also increases (Anselin and Li 2020).
Measures for binary and categorical data
Join count statistic
In addition to measures of spatial autocorrelation of continuous data as seen above, the join count statistic method applies the same concept to binary and categorical data. In essence, the joint count statistic compares the distribution of categorical marks in a lattice with frequencies that would occur randomly. These random occurrences can be computed using a theoretical approximation or random permutations. The same concept was also extended in a multivariate setting with more than two categories. The corresponding spdep functions are called joincount.test and joincount.multi (Dale and Fortin 2014; Bivand 2022; Cliff and Ord 1981).
Implementation using spdep
First, we need to get categorical marks for each data point. We do so by running (non-spatial) PCA on the data
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# Run PCA on the sample
sfe <- runPCA(sfe)
# Cluster based on first 20 PC's and using leiden
colData(sfe)$cluster <- clusterRows(reducedDim(sfe, "PCA")[,1:10],
BLUSPARAM = SNNGraphParam(
cluster.fun = "leiden",
cluster.args = list(
resolution_parameter = 0.3,
objective_function = "modularity")))We can visualise the clusters as follows:
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plotSpatialFeature(sfe,
"cluster",
colGeometryName = colGeometryName
)
The join count statistic is calculated as follows:
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joincount.multi(colData(sfe)$cluster,
colGraph(sfe, colGraphName)) Joincount Expected Variance z-value
1:1 88.57500 22.02256 2.41879 42.7923
2:2 162.18333 72.92803 4.83981 40.5714
3:3 125.58333 52.44683 4.18586 35.7472
4:4 9.02500 1.16112 0.18648 18.2105
2:1 4.52500 80.45865 8.03406 -26.7896
3:1 17.23333 68.24812 7.22423 -18.9802
3:2 34.10000 124.05693 11.82038 -26.1649
4:1 5.16667 10.24812 1.37682 -4.3306
4:2 7.35833 18.62836 1.98087 -8.0075
4:3 12.25000 15.80129 1.82957 -2.6255
Jtot 80.63333 317.44146 16.60077 -58.1209The rows show different combinations of clusters that are in physical contact. E.g. \(1:1\) means the cluster \(1\) with itself. The column Joincount is the observed statistic whereas the column Expected is the expected value of the statistic for this combination. Like this, we can compare whether contacts among cluster combinations occur more frequently than expected at random (Cliff and Ord 1981).
A note of caution
The local methods presented above should always be interpreted with care, since we face the problem of multiple testing when calculating them for each cell. Moreover, the presented methods should mainly serve as exploratory measures to identify interesting regions in the data. Multiple processes can lead to the same pattern, thus the underlying process cannot be inferred from characterising the pattern. Indication of clustering does not explain why this occurs. On one hand, clustering can be the result of spatial interaction between the variables of interest. This is the case if a gene of interest is highly expressed in a tissue region. On the other hand, clustering can be the result of spatial heterogeneity, when local similarity is created by structural heterogeneity in the tissue, e.g., when cells with uniform expression of a gene of interest are grouped together which then creates the apparent clustering of the gene expression measurement.
Session info
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sessionInfo()R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.7
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: Europe/Zurich
tzcode source: internal
attached base packages:
[1] stats4 stats graphics grDevices datasets utils methods
[8] base
other attached packages:
[1] dixon_0.0-8 splancs_2.01-44
[3] sp_2.1-1 bluster_1.10.0
[5] magrittr_2.0.3 stringr_1.5.1
[7] spdep_1.2-8 spData_2.3.0
[9] tmap_3.3-4 scater_1.28.0
[11] scran_1.28.2 scuttle_1.10.3
[13] SFEData_1.2.0 SpatialFeatureExperiment_1.2.3
[15] Voyager_1.2.7 rgeoda_0.0.10-4
[17] digest_0.6.33 sf_1.0-19
[19] reshape2_1.4.4 patchwork_1.3.0
[21] STexampleData_1.8.0 ExperimentHub_2.8.1
[23] AnnotationHub_3.8.0 BiocFileCache_2.8.0
[25] dbplyr_2.3.4 rlang_1.1.4
[27] ggplot2_3.5.1 dplyr_1.1.4
[29] spatstat_3.0-6 spatstat.linnet_3.1-1
[31] spatstat.model_3.2-6 rpart_4.1.19
[33] spatstat.explore_3.3-3 nlme_3.1-162
[35] spatstat.random_3.3-2 spatstat.geom_3.3-4
[37] spatstat.univar_3.1-1 spatstat.data_3.1-4
[39] SpatialExperiment_1.10.0 SingleCellExperiment_1.22.0
[41] SummarizedExperiment_1.30.2 Biobase_2.60.0
[43] GenomicRanges_1.52.1 GenomeInfoDb_1.36.4
[45] IRanges_2.34.1 S4Vectors_0.38.2
[47] BiocGenerics_0.46.0 MatrixGenerics_1.12.3
[49] matrixStats_1.4.1
loaded via a namespace (and not attached):
[1] splines_4.3.1 later_1.3.1
[3] bitops_1.0-9 filelock_1.0.3
[5] tibble_3.2.1 R.oo_1.27.0
[7] polyclip_1.10-7 XML_3.99-0.14
[9] lifecycle_1.0.4 edgeR_3.42.4
[11] lattice_0.21-8 crosstalk_1.2.0
[13] limma_3.56.2 rmarkdown_2.25
[15] yaml_2.3.7 metapod_1.7.0
[17] httpuv_1.6.11 spatstat.sparse_3.1-0
[19] RColorBrewer_1.1-3 DBI_1.2.3
[21] abind_1.4-8 zlibbioc_1.46.0
[23] purrr_1.0.2 R.utils_2.12.3
[25] RCurl_1.98-1.16 rappdirs_0.3.3
[27] GenomeInfoDbData_1.2.10 ggrepel_0.9.4
[29] irlba_2.3.5.1 spatstat.utils_3.1-1
[31] terra_1.7-55 units_0.8-4
[33] goftest_1.2-3 RSpectra_0.16-1
[35] dqrng_0.4.1 DelayedMatrixStats_1.22.6
[37] codetools_0.2-19 DropletUtils_1.20.0
[39] DelayedArray_0.26.7 tidyselect_1.2.1
[41] raster_3.6-26 farver_2.1.2
[43] viridis_0.6.4 ScaledMatrix_1.8.1
[45] base64enc_0.1-3 jsonlite_1.8.9
[47] BiocNeighbors_1.18.0 e1071_1.7-13
[49] ellipsis_0.3.2 dbscan_1.1-11
[51] tools_4.3.1 ggnewscale_0.4.9
[53] Rcpp_1.0.13-1 glue_1.8.0
[55] gridExtra_2.3 xfun_0.40
[57] mgcv_1.9-1 HDF5Array_1.28.1
[59] withr_3.0.2 BiocManager_1.30.22
[61] fastmap_1.2.0 boot_1.3-28.1
[63] rhdf5filters_1.12.1 fansi_1.0.6
[65] rsvd_1.0.5 R6_2.5.1
[67] mime_0.12 colorspace_2.1-1
[69] wk_0.8.0 tensor_1.5
[71] dichromat_2.0-0.1 RSQLite_2.3.8
[73] R.methodsS3_1.8.2 utf8_1.2.4
[75] generics_0.1.3 renv_1.0.3
[77] class_7.3-22 httr_1.4.7
[79] htmlwidgets_1.6.2 S4Arrays_1.0.6
[81] tmaptools_3.1-1 pkgconfig_2.0.3
[83] scico_1.5.0 gtable_0.3.6
[85] blob_1.2.4 XVector_0.40.0
[87] htmltools_0.5.6.1 scales_1.3.0
[89] png_0.1-8 knitr_1.44
[91] rstudioapi_0.15.0 rjson_0.2.23
[93] curl_6.0.1 proxy_0.4-27
[95] cachem_1.1.0 rhdf5_2.44.0
[97] BiocVersion_3.17.1 KernSmooth_2.23-21
[99] vipor_0.4.5 parallel_4.3.1
[101] AnnotationDbi_1.62.2 leafsync_0.1.0
[103] s2_1.1.4 pillar_1.9.0
[105] grid_4.3.1 vctrs_0.6.5
[107] promises_1.2.1 BiocSingular_1.16.0
[109] beachmat_2.16.0 xtable_1.8-4
[111] cluster_2.1.4 beeswarm_0.4.0
[113] evaluate_0.22 magick_2.8.5
[115] cli_3.6.3 locfit_1.5-9.10
[117] compiler_4.3.1 crayon_1.5.3
[119] labeling_0.4.3 classInt_0.4-10
[121] ggbeeswarm_0.7.2 plyr_1.8.9
[123] stringi_1.8.4 stars_0.6-4
[125] viridisLite_0.4.2 deldir_2.0-4
[127] BiocParallel_1.34.2 munsell_0.5.1
[129] Biostrings_2.68.1 leaflet_2.2.0
[131] Matrix_1.5-4.1 leafem_0.2.3
[133] sparseMatrixStats_1.12.2 bit64_4.5.2
[135] Rhdf5lib_1.22.1 statmod_1.5.0
[137] KEGGREST_1.40.1 shiny_1.7.5.1
[139] interactiveDisplayBase_1.38.0 igraph_1.5.1
[141] memoise_2.0.1 lwgeom_0.2-13
[143] bit_4.5.0 ©2024 The pasta authors. Content is published under Creative Commons CC-BY-4.0 License for the text and GPL-3 License for any code.
References
Anselin, Luc. 1995. “Local Indicators of Spatial Association—LISA.” Geographical Analysis 27 (2): 93–115. https://doi.org/10.1111/j.1538-4632.1995.tb00338.x.
———. 2019. “A Local Indicator of Multivariate Spatial Association: Extending Geary’s c.” Geographical Analysis 51 (2): 133–50. https://doi.org/10.1111/gean.12164.
———. 2024. An Introduction to Spatial Data Science with GeoDa: Volume 1: Exploring Spatial Data. CRC Press.
Anselin, Luc, and Xun Li. 2020. “Tobler’s Law in a Multivariate World.” Geographical Analysis 52 (4): 494–510. https://doi.org/https://doi.org/10.1111/gean.12237.
Bivand, Roger. 2022. “R Packages for Analyzing Spatial Data: A Comparative Case Study with Areal Data.” Geographical Analysis 54 (3): 488–518. https://doi.org/10.1111/gean.12319.
Cliff, Andrew David, and J Keith Ord. 1981. Spatial Processes: Models & Applications. Pion, London.
Dale, Mark R. T., and Marie-Josée Fortin. 2014. Spatial Analysis: A Guide for Ecologists. Second Edition. Cambridge ; New York: Cambridge University Press.
Lee, Sang-Il. 2001. “Developing a Bivariate Spatial Association Measure: An Integration of Pearson’s r and Moran’s I.” Journal of Geographical Systems 3 (4): 369–85. https://doi.org/10.1007/s101090100064.
Li, Zhuoxuan, Tianjie Wang, Pentao Liu, and Yuanhua Huang. 2023. “SpatialDM for Rapid Identification of Spatially Co-Expressed Ligand–Receptor and Revealing Cell–Cell Communication Patterns.” Nature Communications 14 (1): 3995. https://doi.org/10.1038/s41467-023-39608-w.
McKellar, David W., Lauren D. Walter, Leo T. Song, Madhav Mantri, Michael F. Z. Wang, Iwijn De Vlaminck, and Benjamin D. Cosgrove. 2021. “Large-Scale Integration of Single-Cell Transcriptomic Data Captures Transitional Progenitor States in Mouse Skeletal Muscle Regeneration.” Communications Biology 4 (1): 1–12. https://doi.org/10.1038/s42003-021-02810-x.
Wartenberg, Daniel. 1985. “Multivariate Spatial Correlation: A Method for Exploratory Geographical Analysis.” Geographical Analysis 17 (4): 263–83. https://doi.org/10.1111/j.1538-4632.1985.tb00849.x.
Zuur, Alain F., Elena N. Ieno, and Graham M. Smith. 2007. Analysing Ecological Data. Statistics for Biology and Health. New York: Springer.